2-Local Isometries on Spaces of Lipschitz Functions

Jiménez-Vargas, A. ; Villegas-Vallecillos, Moisés

Canadian mathematical bulletin, Tome 54 (2011) no. 4, pp. 680-692

Résumé

Let $(X,\,d)$ be a metric space, and let $\text{Lip(}X\text{)}$ denote the Banach space of all scalar-valued bounded Lipschitz functions $f$ on $X$ endowed with one of the natural norms $$\left\| f \right\|\,=\,\max \{{{\left\| f \right\|}_{\infty }},\,L(f)\}\,\,\text{or}\,\,\left\| f \right\|\,=\,{{\left\| f \right\|}_{\infty }}\,+\,L(f),$$ where $L(f)$ is the Lipschitz constant of $f$ . It is said that the isometry group of $\text{Lip(}X\text{)}$ is canonical if every surjective linear isometry of $\text{Lip(}X\text{)}$ is induced by a surjective isometry of $X$ . In this paper we prove that if $X$ is bounded separable and the isometry group of $\text{Lip(}X\text{)}$ is canonical, then every 2-local isometry of $\text{Lip(}X\text{)}$ is a surjective linear isometry. Furthermore, we give a complete description of all 2-local isometries of $\text{Lip(}X\text{)}$ when $X$ is bounded.

Détail

DOI : 10.4153/CMB-2011-025-5

Mots-clés : 46B04, 46J10, 46E15, isometry, local isometry, Lipschitz function

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